Factorial experimental design
Generate an experimental run plan to study several factors together and reveal interactions.
Concepts and reasoning
Learning about factors together
Suppose a process response depends on temperature and mixing speed. Changing temperature at only one speed can miss an interaction: the temperature effect may depend on the speed. A factorial design studies combinations of factor settings so these joint effects can be identified.
For k factors at two levels, a full factorial has 2ᵏ distinct combinations. Code the low and high settings as −1 and +1. The physical low and high values should be feasible, scientifically useful, and sufficiently separated to reveal relevant effects. Coding makes the design balanced and its coefficients easier to compare, but it does not make temperature and speed physically interchangeable.
A main effect is the average response difference between a factor's high and low settings, averaged over the other factors. An interaction means that this difference changes with another factor. When an interaction is important, an isolated statement such as “higher temperature improves yield” can be misleading.
Replication, randomization, and fractions
Randomization distributes time-related disturbances across conditions. Replication supplies independent information about experimental variability. Repeating a measurement on the same experimental unit is not necessarily an independent replicate; distinguish measurement precision from run-to-run variability.
A fractional factorial uses a carefully selected subset of combinations to reduce runs. This saves resources by accepting aliasing: some effects cannot be separated. The choice depends on which interactions can reasonably be assumed small. A fraction is not a full factorial with a few arbitrary rows deleted.
NIST's full factorial introduction and aliasing explanation describe how design structure determines what can be learned. Plan the analysis and plausible interactions before choosing the number of runs.
Mathematical foundations
Effects and regression coefficients
For two coded factors A and B, a model with their interaction is
With balanced −1/+1 coding, the A main effect is 2βₐ and the B main effect is 2βᵦ. The factorial AB interaction effect is 2βₐᵦ. The difference between the two simple A effects, one at each B level, is 4βₐᵦ. These different scalings describe related quantities; label them clearly.
For a regular half fraction of a three-factor design defined by x_C = x_Ax_B, the defining relation is
Multiplying that relation by A gives A = BC, because A² = I for ±1 levels. Similarly, B = AC and C = AB. An estimated A effect therefore combines a main effect with a two-factor interaction. This resolution III design cannot separate those possibilities without extra information or runs.
Worked example
Worked example: an interaction changes the recommendation
Consider hypothetical yield measurements:
| Temperature A | Speed B | Yield |
|---|---|---|
| Low (−1) | Low (−1) | 10 |
| High (+1) | Low (−1) | 14 |
| Low (−1) | High (+1) | 12 |
| High (+1) | High (+1) | 24 |
The average at high A is 19 and at low A is 11, so the A main effect is 8. The B main effect is 18 − 12 = 6. The grand mean is 15.
At low B, increasing A changes yield by 4. At high B it changes yield by 12. Their difference is 8, so βₐᵦ = 8/4 = 2. The fitted model is
Substituting the four settings recovers all four observations. The AB factorial effect is 4, not 8: it compares the mean of the positive-product settings, (10 + 24)/2 = 17, with the negative-product settings, (14 + 12)/2 = 13.
This exact fit does not establish a reliable process model. Four parameters have been estimated from four observations, leaving no residual degrees of freedom to estimate error. Independent replication is needed for a direct error estimate.
For three factors, a full factorial requires eight distinct conditions. Two independent replicates require 16 runs. Adding three center runs gives 19 total runs. Center runs can reveal departure from a linear main-effect surface, but they do not separately identify every factor's quadratic curvature.
Interpreting the results
A design specifies what can be learned
A factorial design varies several factors together so that main effects and interactions can be separated when the design supports them. A main effect averages the high-versus-low contrast over settings of the other factors. An interaction means that one factor's effect changes with another factor's setting; it is not simply a large main effect.
In a fraction, some effects are aliased: the observed contrast cannot distinguish them without additional assumptions or runs. For example, a three-factor half fraction with defining relation I = ABC aliases A with BC. An apparent A effect could therefore reflect A, the BC interaction, or both.
Randomized run order protects against systematic association between settings and time-related disturbances. It does not remove aliasing or estimate effects before responses exist. Interpret the generated plan as an experimental design, not as evidence about which factor matters.
Assumptions and limitations
Define feasible factors and genuine replication
Coded low and high levels must map to meaningful, feasible experimental settings. The interpretation is local to that range; changing a factor from 10 to 20 units is not the same intervention as changing it from 100 to 110. Some combinations may be unsafe or impossible, requiring a constrained design.
Replication means repeating the experimental unit and procedure independently, not merely measuring the same run twice. Repeated measurements quantify measurement error differently from independent process variation. Blocking may be needed when runs span batches, days, or operators.
A fractional design relies on assumptions about which aliased effects are negligible. Higher resolution reduces certain confounding but does not eliminate every ambiguity. Center runs can reveal overall curvature relative to a two-level model, but cannot generally identify each factor's separate quadratic effect.
Diagnostics and sensitivity analysis
Audit the plan before execution
Check the run count, coded-level balance, defining relation, and alias structure against the intended effects. Decide whether an important interaction is confounded with a main effect before collecting data. If it is, consider a larger design or a planned follow-up fraction rather than promising an interpretation the design cannot support.
Translate every coded row into actual operating settings and verify feasibility. Preserve both randomized order and standard order so responses can be joined to the correct design row. Record deviations, batch information, and failed runs.
After execution, analyze response variation, interactions, and residuals using an appropriate fitted model. Center-point disagreement may motivate a response-surface experiment. The NIST factorial-design material introduces how factor combinations determine estimable effects.
References and other material
Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.
- NIST handbook: full factorial designs — Factor combinations and introductory experimental-design concepts.